

where the value \((\widehat{u}^{n+\frac{1}{2}},\widehat{r}^{n+\frac{1}{2}})\) 
can be generated by an extrapolation as suggested in [1]}
or predicted by the first-order scheme (REF ) with half of the time step size:
\(& \frac{\widehat{u}^{n+\frac{1}{2}}-u^n}{{\tau }/2}= \varepsilon ^2\Delta _h \widehat{u}^{n+\frac{1}{2}} - \kappa \widehat{u}^{n+\frac{1}{2}} + \frac{r^n}{\exp \lbrace E_{2h}(u^n)\rbrace }(f(u^n)+\kappa u^n), \\& \ln \widehat{r}^{n+\frac{1}{2}} - \ln r^n= - \frac{r^n}{\exp \lbrace E_{2h}(u^n)\rbrace } \langle f(u^n)+\kappa u^n,\widehat{u}^{n+\frac{1}{2}}-u^n\rangle .\) 
